Companion-Rule-List (CRL): Modal Companion Rules

• Companion-Rule-List (CRL) is a canonical set of modal stable-canonical rules derived from superintuitionistic rule-systems using finite filtration and Gödel translation.
• The CRL framework extends the Blok–Esakia correspondence by mapping si–rules to modal counterparts via finite bounded distributive lattice embeddings.
• Its algorithmic construction and uniform extension to bi-intuitionistic, tense, and polymodal logics offer a semantically transparent method for analyzing rule-system properties.

A Companion-Rule-List (CRL) is a canonical set of modal rules systematically associated to a given superintuitionistic rule-system (si–rule-system) through the machinery of stable (and pre-stable) canonical rules. This association extends the Blok–Esakia Galois connection from logics to the level of rule-systems, offering a uniform, semantically transparent mechanism for capturing modal companions, refutation patterns, and correspondence phenomena across a wide spectrum of logical signatures—including intuitionistic, modal, bi-intuitionistic, and tense logics (Bezhanishvili et al., 2022).

1. Formal Definition and Construction

Given a superintuitionistic rule-system $R$ over some signature $\text{si}$, the Companion-Rule-List $\operatorname{CRL}(R)$ is defined using the following construction:

1. Gödel Translation: For each inference rule $\Gamma/\Delta$ in $R$, apply the Gödel translation $T(\Gamma/\Delta)$ to map formulas into the modal language (standard translation for S4).
2. Modal System Extension: Define $$\tau(R) := S4_R \oplus \{ T(\Gamma/\Delta) : \Gamma/\Delta \in R \}$$, where $S4_R$ is the base modal rule-system.
3. Stable-Canonical Rules: $\operatorname{CRL}(R)$ is then the unique set of modal stable-canonical rules whose universal-class closure equals $\tau(R)$. Each such rule can be written as $\#\langle A\rangle^D$ for a finite S4-algebra $A$ and parameter set $D \subseteq A$:

$$\boxed{ \operatorname{CRL}(R) := \left\{ \#\langle A\rangle^{D} \mid \exists\,(\Gamma/\Delta)\in R\;\; \#\langle A\rangle^{D} \text{ rewrites } \Gamma/\Delta \right\} }$$

Lemma 3.12 ensures that every si-rule $\Gamma/\Delta$ is equivalent (over $IPC_R$) to finitely many stable-canonical rules (Bezhanishvili et al., 2022).

2. Filtration, Embeddings, and Stable-Canonical Blueprints

The methodology behind CRL construction leverages finite filtration and stable-canonical rule encoding:

• Every non-valid si-rule $\Gamma/\Delta$ in a Heyting algebra $H$ is “filtered” to a finite bounded distributive lattice $K$ with an expansion $K'$ yielding a bounded-lattice embedding $K \hookrightarrow H$ that satisfies the Bounded-Domain-Condition (BDC) for the relevant parameters.
• The finite refutation pattern $(K', D)$ is captured by a stable-canonical rule $\#\langle K\rangle^D$:

$$\#\langle K\rangle^D = \left\{ p_{a\wedge b} \leftrightarrow p_a \wedge p_b,\;\; p_{a\vee b} \leftrightarrow p_a \vee p_b,\;\; p_0 \leftrightarrow 0,\;\; p_1 \leftrightarrow 1,\;\; p_{a\to b} \leftrightarrow p_a \to p_b \;\; ((a,b)\in D) \right\} / \left\{ p_a \leftrightarrow p_b : a \ne b \right\}$$

• Propositions 3.10–3.11 establish that $H$ refutes $\#\langle K\rangle^D$ iff there exists a bounded-lattice embedding of $K$ into $H$ satisfying the BDC for $D$; in the dual Esakia-space semantics, this corresponds to a continuous surjection fulfilling the dual BDC (Bezhanishvili et al., 2022).

3. Rule-Level Blok–Esakia Correspondence

A central result is the extension of the Blok–Esakia theorem from logics to rule-systems:

• The maps

$$\tau : \mathbf{Ext}(\operatorname{IPC}_R) \longrightarrow \mathbf{NExt}(\operatorname{S4}_R), \qquad \rho : \mathbf{NExt}(\operatorname{S4}_R) \longrightarrow \mathbf{Ext}(\operatorname{IPC}_R)$$

defined by

$$\tau(R) = S4_R \oplus \{ T(\Gamma/\Delta) : \Gamma/\Delta \in R \}, \quad \rho(M) = \{ \Gamma/\Delta : T(\Gamma/\Delta) \in M \}$$

form a Galois connection, which for $GRZ_R$-extensions is a lattice isomorphism. Specifically,

$$\sigma = \rho^{-1} : \mathbf{Ext}(\operatorname{IPC}_R) \cong \mathbf{NExt}(\operatorname{GRZ}_R) = \tau^{-1}$$

with the property that, for any si–rule–system $R$,

$$\operatorname{Alg}(\tau R) = \tau(\operatorname{Alg}(R)), \quad \operatorname{Alg}(\rho M) = \rho(\operatorname{Alg}(M))$$

Modal companions of $R$ are exactly those $M$ satisfying $\tau(R) \le M \le \sigma(R)$ (Bezhanishvili et al., 2022).

4. Illustrative Examples

Three key cases demonstrate the breadth of the Companion-Rule-List construction and its correspondence properties:

Case	Rule-System	CRL Description
(a) IPC → S4	$IPC_R$ (minimal si)	$\operatorname{CRL}(IPC_R)$ yields S4-canonical rules; reflexivity and transitivity axioms via stable-canonical rules $\#\langle2\rangle$, $\#\langle2\rangle^D$
(b) KM → GL	$KM_R$ (minimal modal-si)	$\operatorname{CRL}(KM_R)$ recovers GL-canonical rules; uses pre-stable canonical rules derived from finite frontons, with each $KM$–rule rewritten into pre-stable rules $\#\langle H\rangle^{(D^{\to}, D^{\boxtimes})}$
(c) Bi-intuitionistic / tense logics	Bi-IPC, Tense signatures	Combine two stable-canonical rules for (→, co→), or for tense (two box modalities); yields an isomorphism $$\mathbf{Ext}(\mathrm{bi\text{-}IPC}_R)\cong\mathbf{NExt}(\mathrm{GRZ.T}_R)$$

Each example follows the uniform recipe: filtration → finite countermodel → canonical rule encoding → aggregation into the CRL.

5. General Recipe and Algorithmic Construction

A constructive “pseudocode” approach realizes $\operatorname{CRL}(R)$ from a finite si–rule–system presentation:

Input: finite-presentation of si-rule-system R Output: finite list CRL(R) of stable-canonical rules CRL ← ∅ for each rule Γ/Δ in R do Θ ← subformula-closure(Γ ∪ Δ) for each finite bounded distributive lattice K generated by Θ do define →_K, … by standard filtered-Heyting construction let D ← { (V(φ), V(ψ)) : φ→ψ ∈ Θ } let rule α ← #⟨K⟩^D if α not already in CRL then add α end for end for return CRL

Mutatis mutandis for modal/fronton cases (requiring Boolean local finiteness).

6. Extensions to Richer Logical Signatures

The Companion-Rule-List methodology extends uniformly to a variety of logical languages and structural enrichments:

• Bi-implication: Add a second filtration parameter for the co-implication connective, extending the stable-canonical rule grammar.
• Tense / Hybrid logics: Introduce two BDC-governed modal parameters (future-box, past-box), generalizing the filtration and encoding process.
• Many-sorted / Polymodal logics: As long as the signature supports locally finite filtration, any non-valid rule admits finite canonicalization into CRL form.

The overarching construction in all cases comprises four steps: select a filtration notion; certify finite countermodels; encode via stable/pres-stable canonical rules; and aggregate the resulting rules to form $\operatorname{CRL}(R)$. This delivers a unified account of rule-level Blok–Esakia, the Dummett–Lemmon conjecture, and the Kuznetsov–Muravitsky isomorphism (Bezhanishvili et al., 2022).

7. Significance and Theoretical Implications

The Companion-Rule-List framework provides a powerful and semantically transparent mechanism for:

• Lifting classical modal–intuitionistic correspondences to the rule-syntactic level.
• Enabling effective axiomatizations in terms of stable and pre-stable canonical rules for any rule-system admitting local finiteness via filtration.
• Clarifying the algebraic and topological dualities inherent in the structure of modal companions, not only for basic intuitionistic and modal logics, but for bi-intuitionistic, tense, and polymodal extensions.
• Systematizing the construction of modal companions and establishing bijective Galois connections in generalized settings, connecting algebraic, syntactic, and semantic perspectives within non-classical logic research (Bezhanishvili et al., 2022).

A plausible implication is that this approach will facilitate further advances in the automated generation of canonical rules for newly proposed logical systems where variations of filtration and stable-canonical construction remain applicable.